User blog:B1mb0w/The S Function (substitution function)
'The S Function (substitution function)' I have written a Version 2 of this function that generalises the function and significantly increases its growth rate. Please refer to that version instead. The S function offers a way to generate very large string sequences (representing large numbers) which grows at a faster rate than \(f_{SVO}(n)\), and may be possible to extend its range to grow at a comparable rate to Ordinal Collapsing Functions. Refer to my The Alpha Function blogs for more information on my work. 'What is the S Function' The S function is a string substitution function, and does not explicitly use any mathematical or transfinite ordinal theory. It is effectively a computer algorithm that can be converted to a program. My next version of my Alpha Function will do this and present the results. The S function is defined recursively as follows: \(S_b^c(a)\) where \(a,b,c\) are S functions 'Definition of the S Function' Starting with: \(S_b^c(a)\) where \(a,b,c\) are S functions We define a valid S function recursively as \(a_m\) where: \(a_0 = S_{b_0 = d_p}^{c_o < n}(n)\) *and *\(d_0 = S_{b_0 < h_q}^{c_o < k_r}(n\) or \(s_n)\) **where **\(h_q\) is a valid S function with this additional restriction that it uses either: ***\(n\) ***smaller subscript \(n\) in \(s_n\) than that used in \(d_0\) **\(k_r\) is a valid S function with this additional restriction \(c_0 = 0\) *then \(a_n\) and \(d_m\) can be constructed using this substitution procedure on \(j_n\): **\(j_{n+1} = S_{b_{n+1} < b_n}^{c_{n+1} < j_n}(j_n)\) Note that \(d_0\) and \(d_p\) are not valid S functions. Therefore a valid S function \(a_m\) will never be equivalent to \(d_0\) or \(d_p\). However \(d_0\) and \(d_p\) are valid components of a valid S function. Therefore they will appear inside a valid S function \(a_m\). Any S Function of the form (\(c = 0)\) collapses to: \(S_b^0(a) = a\) The S Function is familiar with but not identical to FGH Functions. 'Definition of the wildcard character \(s_n\)' A wildcard character \(s_n\) is used to enable further string substitutions in the S Function. There is a hierarchy of \(s_n\) substitution wildcards, starting with \(s_0\). Typically \(s_0\) is substituted for a finite number \(n\). The role of \(s_0\) substitution wildcard, is similar to the role of \(\omega\) in FGH functions. 'An Example of an S Function' An example of a valid S Function \(a_m\) is: * Let \(m = 0, n = 3, b_0 = d_0, c_0 = 1\) \(a_m = a_0 = S_{d_0}^{1}(3) = S_{d_0}(3)\) *Where *\(d_0 = S_{b_0 < h_q}^{c_o < k_r}(n\) or \(s_n)\) *Let *\(n = 0, b_0 = h_0, c_0 = k_0\) \(a_m = S_{S_{h_0}^{k_0}(s_0)}(3)\) *Where *\(h_0\) is a valid S function with this additional restriction that it uses either: **\(n\) **smaller subscript \(n\) in \(s_n\) than that used in \(d_0\) *Let *\(n = 5, c_0 = 0\) \(a_m = S_{S_{5}^{k_0}(s_0)}(3)\) *Where *\(k_0\) is a valid S function with this additional restriction \(c_0 = 0\) *Let *\(n = 1\) \(a_m = S_{S_{5}^{1}(s_0)}(3) = S_{S_{5}(s_0)}(3)\) It will be shown later in this blog that the number of valid S Functions that can be constructed up to this example is greater than \(f_{\zeta_0}(3)\). 'String Substitution Rule-Set' The S function only uses string substitution. Substitutions can occur using one of two functions (or substitution algorithms): Sub() Dec() Rule-set for the Sub() Algorithm: The Sub() algorithm performs string substitutions on one S Function, resulting in equivalent S Functions. For any arbitrary S Function \(S_b^c(a)\) where \(c,b,a\) are any S Functions: Rule-set for the Dec() Algorithm: For any arbitrary S Function \(S()\), the Dec() algorithm involves two steps: 1. Recursively apply the Sub() function on \(S()\) until: \(S() = Sub^{z}(S()) = S_0^{m}(S_b^c(a))\) 2. Decrement \(m\) by 1: \(Dec(S()) = Dec(Sub^{z}(S())) = S_0^{m'}(S_b^c(a))\) where \(m' = Dec(m)\) when \(m\) is an S Function or \(m' = m - 1\) when \(m\) is a finite integer. Rule-set for applying Sub() Algorithm to \(s_{n>0}\) substitution wildcards: The Rule-set for Sub() explains how \(s_0\) wildcards are substituted. The rule-set is extended for all other wildcards as follows: \(Sub(s_{n>0}) = S_{S_{n}^2(s_{n-1}}(s_{n-1})\) 'Growth Rate of the S Function' WORK IN PROGRESS The number of valid S Function sequences that can be constructed has a growth rate faster than \(f_{SVO}(n)\). Here are the growth rates for sub-sets of valid S Function sequences: \(S_{s_1}(n) >> f_{\varphi(\omega,0)}(n)\) \(S_{S_3^2(s_0)}(n) >> f_{\varphi(1,0)}(n) = f_{\epsilon_0}(n)\) \(S_{S_5(s_0)}(n) >> f_{\varphi(2,0)}(n) = f_{\zeta_0}(n)\) \(S_{s_1}(n) >> f_{\varphi(\omega,0)}(n)\) \(S_{s_2}(n) >> f_{\varphi(1,0,0)}(n) = f_{\Gamma_0}(n)\) \(S_{s_m}(n) >> f_{\varphi(1,0_{m})}(n)\) Category:Blog posts